Comment to Manuel

A relation simpler than either Sauveurs or Henflings is the one
Blackwood uses, the relation of the whole tone to the diatonic
semitone. The octave has 5T+2D steps. When d is 0, one gets
5-tet, when T and D are both 1, 7 steps, when 2:1, 12; 3:2, 19;
5:3, 31; 3:1, 17; 4:1, 22 4:3, 26; etc.

"Unrecognizable" diatonic scales result when D where T=1 and D=2, though such as scale might be construed as a
variety of Pelog.

--John


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> but near-octaves will take on the character of equivalence

I can't vouch for having experienced this effect, except through the benefit
of Bill Sethares' 88CET-optimized timbres. Other people experimenting with
88CET have found its off-octaves to be more usable musically than I have. I
pretty much just plain avoid them, but I can imagine how somebody can find use
for them in a few cases. But, except through Bill's mapped timbres, I haven't
heard of the octave-duplicating effect having survived the extremity of detuning
that 88CET imposes. Any perspective from you 88CET experimenters (with
near-harmonic timbres)? Warren, John P., Paul F.?



> Whether this means an increase in harmonic variety, I don't know, because
> (octave) inversions and extensions are very valuable in organizing harmonies
> into coherent progressions. Perhaps it's an increase in variety, but at the
> expense of unity.

My composition entitled "Night and Day" (which has received mixed reviews),
pitted very traditional harmony in 88CET against its more native nontraditional
resources. The traditional harmony was exceptionally difficult to realize
because only a handful of the voicings that you'd expect to be freely available
through octave-displacement, were actually available.


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>> but near-octaves will take on the character of equivalence

>I can't vouch for having experienced this effect, except through the
>benefit
>of Bill Sethares' 88CET-optimized timbres. Other people experimenting with
>88CET have found its off-octaves to be more usable musically than I have.
I
>pretty much just plain avoid them, but I can imagine how somebody can find
use
>for them in a few cases. But, except through Bill's mapped timbres, I
haven't
>heard of the octave-duplicating effect having survived the extremity of
>detuning
>that 88CET imposes. Any perspective from you 88CET experimenters (with
>near-harmonic timbres)? Warren, John P., Paul F.?

I was not referring to 88CET, as I have never worked with this tuning. Nor
was I saying that the nearest available approximation to an octave would
automatically work like one. The point is that if a note comes close enough
to an octave or a multiple octave, it will sound equivalent, especially in
the case of harmonic partials. For example, an interval of 33 Pierce steps
exhibits equivalence, even though it is a very different pitch class in the
tritave scheme. Even when the even partials are removed, I believe the
virtual pitch sensation is not very octave-specific.

In the case of inharmonic partials, octave equivalence may play less of a
role, but still exists, and is less demanding as to intonation. Gamelan
music is a case in point: a recent post indicated that Javanese musicians
are well aware of the pure octave, probably through second-order beating,
but they prefer to detune it and find quite a wide range of acceptable
octave sizes.


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> The point is that if a note comes close enough
> to an octave or a multiple octave, it will sound equivalent, especially in
> the case of harmonic partials. For example, an interval of 33 Pierce steps
> exhibits equivalence, even though it is a very different pitch class in the
> tritave scheme.

Based upon my 88CET experience at least, I have found that they sound like
different pitch classes in the usual octave scheme as well. I haven't tried 33
Pierce-Bohlen steps for comparison.

I personally find that, despite classical theoretical premises,
octave-equivalence weakens with the more octaves you stack up. The feeling, to
my ears at least, that two Cs three or four octaves apart are of the same pitch
class, is less compelling than two Cs a single octave apart.

That of course depends heavily on how the two notes are used. Especially,
whether they're played together or in sequence, and how many notes appear
between them. And of course it depends upon timbre.

Based upon that, I suspect that the (roughly) eighth-tone error of a
33-Pierce-Bohlen-step interval would be less ... uhmmm ... "damaging" for lack
of a better word ... to the four-octave sensation than that (roughly) same error
from 88CET's closest approximation to a single octave, because the four-octave
sensation is comparatively weak to start with.


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